Finite Group Databasebeta

← #33005136

Subgroups — #33005136

Order 18688.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988412C211yesmaximal, characteristic, complemented
2C2×#17988388C211yesmaximal, characteristic, complemented
3C2×#17988388C211yesmaximal, characteristic, complemented
4C2×C22⋊#11924003C211yesmaximal, characteristic, complemented
5#22609960C211yesmaximal, characteristic, complemented
6#22608022C211yesmaximal, characteristic, complemented
7C2×#466×C73C211yesmaximal, characteristic
8C22×#11924003C2211yescharacteristic
9C22×#11924003C2211yescharacteristic
10C2×#14556525C2211yescharacteristic
11C2×#14556525C2211yescharacteristic
12#17988447—21no—
13#17988412—22no—
14#17988388—22no—
15#17988388—22no—
16C22⋊#11924003—22no—
17#17988350—21no—
18C22×D8×C73C2211yescharacteristic
19C23×C8×C73C2211yescharacteristic
20C2×#121×C73—21no—
21#466×C73—22no—
22#466×C73—22no—
23C2×#107×C73C2211yescharacteristic
24C2×#102×C73—21no—
25C2×C4⋊Dic73—21no—
26C2×C4⋊Dic73—21no—
27C2×C22⋊Dic73—41no—
28C2×#11924003—21no—
29C2×#11924003—21no—
30C2×#11924003—21no—
31C2×#11924003—21no—
32C2×#11924003—21no—
33C2×#11924003D412yescomplemented
34#14556525—21no—
35#14556525—21no—
36#14556525—21no—
37#14556525—21no—
38#14556498—41no—
39C22×D4×C73—21no—
40C22×D4×C73—41no—
41C23×C4×C73D411yescharacteristic
42C2×D8×C73—21no—
43C2×D8×C73—21no—
44C2×D8×C73—21no—
45C2×D8×C73—21no—
46C2×D8×C73—21no—
47C2×D8×C73D412yes—
48C22×C8×C73—21no—
49C22×C8×C73—21no—
50C22×C8×C73—21no—
51C22×C8×C73—21no—
52C22×C8×C73—21no—
53C22×C8×C73—21no—
54C22×C8×C73D411yescharacteristic
55C22×C8×C73C2311yescharacteristic
56#121×C73—24no—
57#121×C73—42no—
58C2×C4⋊C4×C73—21no—
59C2×C22⋊C4×C73—41no—
60#107×C73—21no—
61#107×C73D412yes—
62#102×C73—24no—
63C22×Dic73—41no—
64C22×Dic73—41no—
65C4⋊Dic73—41no—
66C4⋊Dic73—41no—
67C4⋊Dic73—41no—
68C4⋊Dic73—41no—
69C22⋊Dic73—82no—
70#11924003—22no—
71#11924003—41no—
72C24×C73—41no—
73C24×C73D811yescharacteristic
74C2×D4×C73—22no—
75C2×D4×C73—44no—
76C2×D4×C73—41no—
77C2×D4×C73—41no—
78C2×D4×C73—41no—
79C2×D4×C73—41no—
80C2×D4×C73—41no—
81C2×D4×C73—82no—
82C2×D4×C73—82no—
83C22×C4×C73—21no—
84C22×C4×C73—21no—
85C22×C4×C73—21no—
86C22×C4×C73—21no—
87C22×C4×C73—21no—
88C22×C4×C73—21no—
89C22×C4×C73—41no—
90C22×C4×C73D811yescharacteristic
91C22×C4×C73C2×D411yescharacteristic
92D8×C73—22no—
93D8×C73—22no—
94D8×C73—41no—
95D8×C73—41no—
96C2×C8×C73—21no—
97C2×C8×C73—21no—
98C2×C8×C73—21no—
99C2×C8×C73—41no—
100C2×C8×C73—41no—
101C2×C8×C73—41no—
102C2×C8×C73—41no—
103C2×C8×C73C2×D411yescharacteristic
104C2×C8×C73C2×D411yescharacteristic
105C2×C8×C73C2×D411yescharacteristic
106C2×C8×C73#4111yescharacteristic
107C2×C8×C73#4111yescharacteristic
108C2×C8×C73#4111yescharacteristic
109C4⋊C4×C73—22no—
110C4⋊C4×C73—41no—
111C22⋊C4×C73—44no—
112C23×C73—21no—
113C23×C73—21no—
114C23×C73—21no—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73—21no—
118C23×C73—42no—
119C23×C73—81no—
120C23×C73—81no—
121C23×C73—81no—
122C23×C73—81no—
123C23×C73—81no—
124C23×C73—81no—
125C23×C73D1612yes—
126C23×C73C2×D811yescharacteristic
127D4×C73—42no—
128D4×C73—42no—
129D4×C73—88no—
130D4×C73—81no—
131D4×C73—81no—
132C2×C4×C73—21no—
133C2×C4×C73—21no—
134C2×C4×C73—21no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—42no—
140C2×C4×C73—81no—
141C2×C4×C73—81no—
142C2×C4×C73#12011yescharacteristic
143C2×C4×C73#12111yescharacteristic
144C2×C4×C73#12311yescharacteristic
145C2×C4×C73#13511yescharacteristic
146C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C73#13711yescharacteristic
148C2×Dic73—81no—
149C2×Dic73—81no—
150C2×Dic73—81no—
151C2×Dic73—81no—
152C2×Dic73—81no—
153C2×Dic73—81no—
154C8×C73—41no—
155C8×C73#12111yescharacteristic
156C8×C73#12111yescharacteristic
157C8×C73#12311yescharacteristic
158C8×C73#12411yescharacteristic
159C22×C73—22no—
160C22×C73—22no—
161C22×C73—22no—
162C22×C73—41no—
163C22×C73—41no—
164C22×C73—41no—
165C22×C73—41no—
166C22×C73—82no—
167C22×C73—82no—
168C22×C73—82no—
169C22×C73—161no—
170C22×C73—161no—
171C22×C73—161no—
172C22×C73—161no—
173C22×C73#44711yescharacteristic
174C22×C73#46611yescharacteristic
175C22×C73#48011yescharacteristic
176C22×C73C2×D1611yescharacteristic
177C22×C73#50811yescharacteristic
178C22×C73#50911yescharacteristic
179C22×C73#51011yescharacteristic
180C4×C73—41no—
181C4×C73—82no—
182C4×C73#39411yescharacteristic
183C4×C73#45211yescharacteristic
184C4×C73#46811yescharacteristic
185C4×C73#48211yescharacteristic
186Dic73—161no—
187Dic73—161no—
188#13380—731nomaximal, Sylow
189C2×C73—42no—
190C2×C73—162no—
191C2×C73#203311yescharacteristic
192C2×C73#218411yescharacteristic
193C2×C73#218911yescharacteristic
194C2×C73#221511yescharacteristic
195C2×C73#222011yescharacteristic
196C2×C73#223211yescharacteristic
197C2×C73#223411yescharacteristic
198C2×#466D7311yescharacteristic
199C2×#366—731no—
200C2×#357—731no—
201C2×#357—731no—
202C2×C22⋊C16—731no—
203#1808—731no—
204#1380—731no—
205C73#1338011yescharacteristic, complemented, Sylow
206C22×D8C2×D7311yescharacteristic
207C23×C8C2×D7311yescharacteristic
208C2×#121—21no—
209C22×C16—731no—
210C22×C16—731no—
211#466—22no—
212#466—22no—
213C2×#107—731no—
214C2×#107—731no—
215C2×#107C2×D7311yescharacteristic
216C2×#102—21no—
217#380—1461no—
218#366—1462no—
219#357—1462no—
220#357—1462no—
221C22⋊C16—1462no—
222#340—1461no—
223C22×D4—21no—
224C22×D4—41no—
225C23×C4C22⋊D7311yescharacteristic
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—21no—
229C2×D8—21no—
230C2×D8—21no—
231C2×D8C22⋊D7312yes—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8C22⋊D7311yescharacteristic
239C22×C8C22×D7311yescharacteristic
240#121—24no—
241#121—42no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—1461no—
244C2×C4⋊C4—1461no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—2921no—
247C2×C16—732no—
248C2×C16—1461no—
249C2×C16—1461no—
250C2×C16—1461no—
251C2×C16—1461no—
252C2×C16—1461no—
253#107—21no—
254#107—1461no—
255#107—1461no—
256#107—1461no—
257#107—1461no—
258#107C22⋊D7312yes—
259#102—24no—
260#95—2921no—
261C24—41no—
262C24#1192401311yescharacteristic
263C2×D4—22no—
264C2×D4—44no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—82no—
271C2×D4—82no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—2921no—
280C22×C4—2921no—
281C22×C4#1192401311yescharacteristic
282C22×C4C2×C22⋊D7311yescharacteristic
283D8—22no—
284D8—22no—
285D8—41no—
286D8—41no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8D4×D7311yescharacteristic
295C2×C8C2×C22⋊D7311yescharacteristic
296C2×C8C2×C22⋊D7311yescharacteristic
297C2×C8#1192402711yescharacteristic
298C2×C8#1192402711yescharacteristic
299C2×C8#1192402711yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—2921no—
303C4⋊C4—2921no—
304C4⋊C4—2921no—
305C4⋊C4—2921no—
306C22⋊C4—44no—
307C22⋊C4—5842no—
308C16—2921no—
309C16—1462no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—42no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23#1455653212yes—
324C23C2×#1192401311yescharacteristic
325D4—42no—
326D4—42no—
327D4—88no—
328D4—81no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—5841no—
341C2×C4—5841no—
342C2×C4—5841no—
343C2×C4—5841no—
344C2×C4—5841no—
345C2×C4—5841no—
346C2×C4#1455656011yescharacteristic
347C2×C4#1455656311yescharacteristic
348C2×C4#1455657211yescharacteristic
349C2×C4#1455661111yescharacteristic
350C2×C4#1455661711yescharacteristic
351C2×C4#1455662111yescharacteristic
352C8—41no—
353C8#1455656211yescharacteristic
354C8#1455656311yescharacteristic
355C8#1455657211yescharacteristic
356C8#1455657811yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—161no—
368C22—161no—
369C22—161no—
370C22—161no—
371C22#1798872111yescharacteristic
372C22#1798882111yescharacteristic
373C22#1798889811yescharacteristic
374C22C2×#1455653211yescharacteristic
375C22#1798901711yescharacteristic
376C22#1798902311yescharacteristic
377C22#1798902711yescharacteristic
378C4—41no—
379C4—82no—
380C4—11681no—
381C4—11681no—
382C4#1798850111yescharacteristic
383C4#1798874911yescharacteristic
384C4#1798883111yescharacteristic
385C4#1798890811yescharacteristic
386C2—42no—
387C2—162no—
388C2#2261111011yescharacteristic
389C2#2261184311yescharacteristic
390C2#2261187111yescharacteristic
391C2#2261201511yescharacteristic
392C2#2261204211yescharacteristic
393C2#2261210911yescharacteristic
394C2#2261211911yescharacteristic
subgroupG/H
centerC23C2×#11924013
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
Fitting, prime radicalC2×#466×C73C2
socleC23×C73C2×D8
hypercenterC2×#466D73
hypocenterC73#13380
LCSG ▷ C22×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#17988412C2semidirect
2. C2×#17988388C2semidirect
3. C2×#17988388C2semidirect
4. C2×C22⋊#11924003C2semidirect
5. #22609960C2semidirect
6. #22608022C2semidirect
33. C2×#11924003D4semidirect
205. C73#13380semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179884122
C2×#179883882
C2×C22⋊#119240032
#226099602
#226080222
C2×#466×C732
#1338073